Probabilistic phylogenetic inference with insertions and deletions

A fundamental task in sequence analysis is to calculate the probability of a multiple alignment given a phylogenetic tree relating the sequences and an evolutionary model describing how sequences change over time. However, the most widely used phylogenetic models only account for residue substitution events. We describe a probabilistic model of a multiple sequence alignment that accounts for insertion and deletion events in addition to substitutions, given a phylogenetic tree, using a rate matrix augmented by the gap character. Starting from a continuous Markov process, we construct a non-reversible generative (birth-death) evolutionary model for insertions and deletions. The model assumes that insertion and deletion events occur one residue at a time. We apply this model to phylogenetic tree inference by extending the program dnaml in phylip. Using standard benchmarking methods on simulated data and a new "concordance test" benchmark on real ribosomal RNA alignments, we show that the extended program dnamlepsilon improves accuracy relative to the usual approach of ignoring gaps, while retaining the computational efficiency of the Felsenstein peeling algorithm.     

Sequence alignments and pair hidden Markov models using evolutionary history

This work presents a novel pairwise statistical alignment method based on an explicit evolutionary model of insertions and deletions (indels). Indel events of any length are possible according to a geometric distribution. The geometric distribution parameter, the indel rate, and the evolutionary time are all maximum likelihood estimated from the sequences being aligned. Probability calculations are done using a pair hidden Markov model (HMM) with transition probabilities calculated from the indel parameters. Equations for the transition probabilities make the pair HMM closely approximate the specified indel model. The method provides an optimal alignment, its likelihood, the likelihood of all possible alignments, and the reliability of individual alignment regions. Human alpha and beta-hemoglobin sequences are aligned, as an illustration of the potential utility of this pair HMM approach.     

Joint Bayesian estimation of alignment and phylogeny

We describe a novel model and algorithm for simultaneously estimating multiple molecular sequence alignments and the phylogenetic trees that relate the sequences. Unlike current techniques that base phylogeny estimates on a single estimate of the alignment, we take alignment uncertainty into account by considering all possible alignments. Furthermore, because the alignment and phylogeny are constructed simultaneously, a guide tree is not needed. This sidesteps the problem in which alignments created by progressive alignment are biased toward the guide tree used to generate them. Joint estimation also allows us to model rate variation between sites when estimating the alignment and to use the evidence in shared insertion/deletions (indels) to group sister taxa in the phylogeny. Our indel model makes use of affine gap penalties and considers indels of multiple letters. We make the simplifying assumption that the indel process is identical on all branches. As a result, the probability of a gap is independent of branch length. We use a Markov chain Monte Carlo (MCMC) method to sample from the posterior of the joint model, estimating the most probable alignment and tree and their support simultaneously. We describe a new MCMC transition kernel that improves our algorithm's mixing efficiency, allowing the MCMC chains to converge even when started from arbitrary alignments. Our software implementation can estimate alignment uncertainty and we describe a method for summarizing this uncertainty in a single plot.     

Tree-based maximal likelihood substitution matrices and hidden Markov models

There has been considerable interest in the problem of making maximum likelihood (ML) evolutionary trees which allow insertions and deletions. This problem is partly one of formulation: how does one define a probabilistic model for such trees which treats insertion and deletion in a biologically plausible manner? A possible answer to this question is proposed here by extending the concept of a hidden Markov model (HMM) to evolutionary trees. The model, called a tree-HMM, allows what may be loosely regarded as learnable affine-type gap penalties for alignments. These penalties are expressed in HMMs as probabilities of transitions between states. In the tree-HMM, this idea is given an evolutionary embodiment by defining trees of transitions. Just as the probability of a tree composed of ungapped sequences is computed, by Felsenstein's method, using matrices representing the probabilities of substitutions of residues along the edges of the tree, so the probabilities in a tree-HMM are computed by substitution matrices for both residues and transitions. How to define these matrices by a ML procedure using an algorithm that learns from a database of protein sequences is shown here. Given these matrices, one can define a tree-HMM likelihood for a set of sequences, assuming a particular tree topology and an alignment of the sequences to the model. If one could efficiently find the alignment which maximizes (or comes close to maximizing) this likelihood, then one could search for the optimal tree topology for the sequences. An alignment algorithm is defined here which, given a particular tree topology, is guaranteed to increase the likelihood of the model. Unfortunately, it fails to find global optima for realistic sequence sets. Thus further research is needed to turn the tree-HMM into a practical phylogenetic tool.     

A novel approach to remote homology detection: jumping alignments.

We describe a new algorithm for protein classification and the detection of remote homologs. The rationale is to exploit both vertical and horizontal information of a multiple alignment in a well-balanced manner. This is in contrast to established methods such as profiles and profile hidden Markov models which focus on vertical information as they model the columns of the alignment independently and to family pairwise search which focuses on horizontal information as it treats given sequences separately. In our setting, we want to select from a given database of "candidate sequences" those proteins that belong to a given superfamily. In order to do so, each candidate sequence is separately tested against a multiple alignment of the known members of the superfamily by means of a new jumping alignment algorithm. This algorithm is an extension of the Smith-Waterman algorithm and computes a local alignment of a single sequence and a multiple alignment. In contrast to traditional methods, however, this alignment is not based on a summary of the individual columns of the multiple alignment. Rather, the candidate sequence is at each position aligned to one sequence of the multiple alignment, called the "reference sequence." In addition, the reference sequence may change within the alignment, while each such jump is penalized. To evaluate the discriminative quality of the jumping alignment algorithm, we compare it to profiles, profile hidden Markov models, and family pairwise search on a subset of the SCOP database of protein domains. The discriminative quality is assessed by median false positive counts (med-FP-counts). For moderate med-FP-counts, the number of successful searches with our method is considerably higher than with the competing methods.     

Evolutionary Triplet Models of Structured RNA

The reconstruction and synthesis of ancestral RNAs is a feasible goal for paleogenetics. This will require new bioinformatics methods, including a robust statistical framework for reconstructing histories of substitutions, indels and structural changes. We describe a “transducer composition” algorithm for extending pairwise probabilistic models of RNA structural evolution to models of multiple sequences related by a phylogenetic tree. This algorithm draws on formal models of computational linguistics as well as the 1985 protosequence algorithm of David Sankoff. The output of the composition algorithm is a multiple-sequence stochastic context-free grammar. We describe dynamic programming algorithms, which are robust to null cycles and empty bifurcations, for parsing this grammar. Example applications include structural alignment of non-coding RNAs, propagation of structural information from an experimentally-characterized sequence to its homologs, and inference of the ancestral structure of a set of diverged RNAs. We implemented the above algorithms for a simple model of pairwise RNA structural evolution; in particular, the algorithms for maximum likelihood (ML) alignment of three known RNA structures and a known phylogeny and inference of the common ancestral structure. We compared this ML algorithm to a variety of related, but simpler, techniques, including ML alignment algorithms for simpler models that omitted various aspects of the full model and also a posterior-decoding alignment algorithm for one of the simpler models. In our tests, incorporation of basepair structure was the most important factor for accurate alignment inference; appropriate use of posterior-decoding was next; and fine details of the model were least important. Posterior-decoding heuristics can be substantially faster than exact phylogenetic inference, so this motivates the use of sum-over-pairs heuristics where possible (and approximate sum-over-pairs). For more exact probabilistic inference, we discuss the use of transducer composition for ML (or MCMC) inference on phylogenies, including possible ways to make the core operations tractable.     

Markov-modulated Markov chains and the covarion process of molecular evolution.

The covarion (or site specific rate variation, SSRV) process of biological sequence evolution is a process by which the evolutionary rate of a nucleotide/amino acid/codon position can change in time. In this paper, we introduce time-continuous, space-discrete, Markov-modulated Markov chains as a model for representing SSRV processes, generalizing existing theory to any model of rate change. We propose a fast algorithm for diagonalizing the generator matrix of relevant Markov-modulated Markov processes. This algorithm makes phylogeny likelihood calculation tractable even for a large number of rate classes and a large number of states, so that SSRV models become applicable to amino acid or codon sequence datasets. Using this algorithm, we investigate the accuracy of the discrete approximation to the Gamma distribution of evolutionary rates, widely used in molecular phylogeny. We show that a relatively large number of classes is required to achieve accurate approximation of the exact likelihood when the number of analyzed sequences exceeds 20, both under the SSRV and among site rate variation (ASRV) models.     

Indel information eliminates trivial sequence alignment in maximum likelihood phylogenetic analysis

Although there has been a recent proliferation in maximum‐likelihood (ML)‐based tree estimation methods based on a fixed sequence alignment (MSA), little research has been done on incorporating indel information in this traditional framework. We show, using a simple model on a single character example, that a trivial alignment of a different form than that previously identified for parsimony is optimal in ML under standard assumptions treating indels as “missing” data, but that it is not optimal when indels are incorporated into the character alphabet. We show that the optimality of the trivial alignment is not an artefact of simplified theory assumptions by demonstrating that trivial alignment likelihoods of five different multiple sequence alignment datasets exhibit this phenomenon. These results demonstrate the need for use of indel information in likelihood analysis on fixed MSAs, and suggest that caution must be exercised when drawing conclusions from software implementations claiming improvements in likelihood scores under an indels‐as‐missing assumption. © The Willi Hennig Society 2012.     

Statistical significance of probabilistic sequence alignment and related local hidden Markov models.

The score statistics of probabilistic gapped local alignment of random sequences is investigated both analytically and numerically. The full probabilistic algorithm (e.g., the "local" version of maximum-likelihood or hidden Markov model method) is found to have anomalous statistics. A modified "semi-probabilistic" alignment consisting of a hybrid of Smith-Waterman and probabilistic alignment is then proposed and studied in detail. It is predicted that the score statistics of the hybrid algorithm is of the Gumbel universal form, with the key Gumbel parameter lambda taking on a fixed asymptotic value for a wide variety of scoring systems and parameters. A simple recipe for the computation of the "relative entropy," and from it the finite size correction to lambda, is also given. These predictions compare well with direct numerical simulations for sequences of lengths between 100 and 1,000 examined using various PAM substitution scores and affine gap functions. The sensitivity of the hybrid method in the detection of sequence homology is also studied using correlated sequences generated from toy mutation models. It is found to be comparable to that of the Smith-Waterman alignment and significantly better than the Viterbi version of the probabilistic alignment.     

Positional statistical significance in sequence alignment.

Beginning with the concept of near-optimal sequence alignments, we can assign a probability that each element in one sequence is paired in an alignment with each element in another sequence. This involves a sum over the set of all possible pairwise alignments. The method employs a designed hidden Markov model (HMM) and the rigorous forward and forward-backward algorithms of Rabiner. The approach can use any standard sequence-element-to-element probabilistic similarity measures and affine gap penalty functions. This allows the positional alignment statistical significance to be obtained as a function of such variables. A measure of the probabilistic relationship between any single sequence and a set of sequences can be directly obtained. In addition, the employed HMM with the Viterbi algorithm provides a simple link to the standard dynamic programming optimal alignment algorithms.     

Bayesian segmental models with multiple sequence alignment profiles for protein secondary structure and contact map prediction

In this paper, we develop a segmental semi-Markov model (SSMM) for protein secondary structure prediction which incorporates multiple sequence alignment profiles with the purpose of improving the predictive performance. The segmental model is a generalization of the hidden Markov model where a hidden state generates segments of various length and secondary structure type. A novel parameterized model is proposed for the likelihood function that explicitly represents multiple sequence alignment profiles to capture the segmental conformation. Numerical results on benchmark data sets show that incorporating the profiles results in substantial improvements and the generalization performance is promising. By incorporating the information from long range interactions in /spl beta/-sheets, this model is also capable of carrying out inference on contact maps. This is an important advantage of probabilistic generative models over the traditional discriminative approach to protein secondary structure prediction. The Web server of our algorithm and supplementary materials are available at http://public.kgi.edu/-wild/bsm.html.     

OPTIMIZATION ALIGNMENT: THE END OF MULTIPLE SEQUENCE ALIGNMENT IN PHYLOGENETICS?

Abstract— A method is described to assess directly the number of DNA sequence transformations, evolutionary events, required by a phylogenetic topology without the use of multiple sequence alignment. This is accomplished through a generalization of existing character optimization procedures to include insertion and deletion events (indels) in addition to base substitutions. The crux of the model is the treatment of indels as processes as opposed to the patterns implied by multiple sequence alignment. The results of this procedure are directly compatible with parsimony-based tree lengths. In addition to the simplicity of the method, it appears to generate more efficient (simpler) explanations of sequence variation than does multiple alignment.     

Statistical Alignment with a Sequence Evolution Model Allowing Rate Heterogeneity along the Sequence

We present a stochastic sequence evolution model to obtain alignments and estimate mutation rates between two homologous sequences. The model allows two possible evolutionary behaviors along a DNA sequence in order to determine conserved regions and take its heterogeneity into account. In our model, the sequence is divided into slow and fast evolution regions. The boundaries between these sections are not known. It is our aim to detect them. The evolution model is based on a fragment insertion and deletion process working on fast regions only and on a substitution process working on fast and slow regions with different rates. This model induces a pair hidden Markov structure at the level of alignments, thus making efficient statistical alignment algorithms possible. We propose two complementary estimation methods, namely, a Gibbs sampler for Bayesian estimation and a stochastic version of the EM algorithm for maximum likelihood estimation. Both algorithms involve the sampling of alignments. We propose a partial alignment sampler, which is computationally less expensive than the typical whole alignment sampler. We show the convergence of the two estimation algorithms when used with this partial sampler. Our algorithms provide consistent estimates for the mutation rates and plausible alignments and sequence segmentations on both simulated and real data.     

Picking alignments from (Steiner) trees.

The application of Needleman-Wunsch alignment techniques to biological sequences is complicated by two serious problems when the sequences are long: the running time, which scales as the product of the lengths of sequences, and the difficulty in obtaining suitable parameters that produce meaningful alignments. The running time problem is often corrected by reducing the search space, using techniques such as banding, or chaining of high-scoring pairs. The parameter problem is more difficult to fix, partly because the probabilistic model, which Needleman-Wunsch is equivalent to, does not capture a key feature of biological sequence alignments, namely the alternation of conserved blocks and seemingly unrelated nonconserved segments. We present a solution to the problem of designing efficient search spaces for pair hidden Markov models that align biological sequences by taking advantage of their associated features. Our approach leads to an optimization problem, for which we obtain a 2-approximation algorithm, and that is based on the construction of Manhattan networks, which are close relatives of Steiner trees. We describe the underlying theory and show how our methods can be applied to alignment of DNA sequences in practice, successfully reducing the Viterbi algorithm search space of alignment PHMMs by three orders of magnitude.     

MUMMALS: multiple sequence alignment improved by using hidden Markov models with local structural information

We have developed MUMMALS, a program to construct multiple protein sequence alignment using probabilistic consistency. MUMMALS improves alignment quality by using pairwise alignment hidden Markov models (HMMs) with multiple match states that describe local structural information without exploiting explicit structure predictions. Parameters for such models have been estimated from a large library of structure-based alignments. We show that (i) on remote homologs, MUMMALS achieves statistically best accuracy among several leading aligners, such as ProbCons, MAFFT and MUSCLE, albeit the average improvement is small, in the order of several percent; (ii) a large collection (>10000) of automatically computed pairwise structure alignments of divergent protein domains is superior to smaller but carefully curated datasets for estimation of alignment parameters and performance tests; (iii) reference-independent evaluation of alignment quality using sequence alignment-dependent structure superpositions correlates well with reference-dependent evaluation that compares sequence-based alignments to structure-based reference alignments.     

Probabilistic alignments with quality scores: an application to short-read mapping toward accurate SNP/indel detection

MOTIVATION: Recent studies have revealed the importance of considering quality scores of reads generated by next-generation sequence (NGS) platforms in various downstream analyses. It is also known that probabilistic alignments based on marginal probabilities (e.g. aligned-column and/or gap probabilities) provide more accurate alignment than conventional maximum score-based alignment. There exists, however, no study about probabilistic alignment that considers quality scores explicitly, although the method is expected to be useful in SNP/indel callers and bisulfite mapping, because accurate estimation of aligned columns or gaps is important in those analyses. RESULTS: In this study, we propose methods of probabilistic alignment that consider quality scores of (one of) the sequences as well as a usual score matrix. The method is based on posterior decoding techniques in which various marginal probabilities are computed from a probabilistic model of alignments with quality scores, and can arbitrarily trade-off sensitivity and positive predictive value (PPV) of prediction (aligned columns and gaps). The method is directly applicable to read mapping (alignment) toward accurate detection of SNPs and indels. Several computational experiments indicated that probabilistic alignments can estimate aligned columns and gaps accurately, compared with other mapping algorithms e.g. SHRiMP2, Stampy, BWA and Novoalign. The study also suggested that our approach yields favorable precision for SNP/indel calling.     

Evolutionary HMMs: a Bayesian approach to multiple alignment

MOTIVATION: We review proposed syntheses of probabilistic sequence alignment, profiling and phylogeny. We develop a multiple alignment algorithm for Bayesian inference in the links model proposed by Thorne et al. (1991, J. Mol. Evol., 33, 114-124). The algorithm, described in detail in Section 3, samples from and/or maximizes the posterior distribution over multiple alignments for any number of DNA or protein sequences, conditioned on a phylogenetic tree. The individual sampling and maximization steps of the algorithm require no more computational resources than pairwise alignment. METHODS: We present a software implementation (Handel) of our algorithm and report test results on (i) simulated data sets and (ii) the structurally informed protein alignments of BAliBASE (Thompson et al., 1999, Nucleic Acids Res., 27, 2682-2690). RESULTS: We find that the mean sum-of-pairs score (a measure of residue-pair correspondence) for the BAliBASE alignments is only 13% lower for Handelthan for CLUSTALW(Thompson et al., 1994, Nucleic Acids Res., 22, 4673-4680), despite the relative simplicity of the links model (CLUSTALW uses affine gap scores and increased penalties for indels in hydrophobic regions). With reference to these benchmarks, we discuss potential improvements to the links model and implications for Bayesian multiple alignment and phylogenetic profiling. AVAILABILITY: The source code to Handelis freely distributed on the Internet at http://www.biowiki.org/Handel under the terms of the GNU Public License (GPL, 2000, http://www.fsf.org./copyleft/gpl.html).     

indel-Seq-Gen: a new protein family simulator incorporating domains, motifs, and indels

Reconstructing the evolutionary history of protein sequences will provide a better understanding of divergence mechanisms of protein superfamilies and their functions. Long-term protein evolution often includes dynamic changes such as insertion, deletion, and domain shuffling. Such dynamic changes make reconstructing protein sequence evolution difficult and affect the accuracy of molecular evolutionary methods, such as multiple alignments and phylogenetic methods. Unfortunately, currently available simulation methods are not sufficiently flexible and do not allow biologically realistic dynamic protein sequence evolution. We introduce a new method, indel-Seq-Gen (iSG), that can simulate realistic evolutionary processes of protein sequences with insertions and deletions (indels). Unlike other simulation methods, iSG allows the user to simulate multiple subsequences according to different evolutionary parameters, which is necessary for generating realistic protein families with multiple domains. iSG tracks all evolutionary events including indels and outputs the "true" multiple alignment of the simulated sequences. iSG can also generate a larger sequence space by allowing the use of multiple related root sequences. With all these functions, iSG can be used to test the accuracy of, for example, multiple alignment methods, phylogenetic methods, evolutionary hypotheses, ancestral protein reconstruction methods, and protein family classification methods. We empirically evaluated the performance of iSG against currently available methods by simulating the evolution of the G protein-coupled receptor and lipocalin protein families. We examined their true multiple alignments, reconstruction of the transmembrane regions and beta-strands, and the results of similarity search against a protein database using the simulated sequences. We also presented an example of using iSG for examining how phylogenetic reconstruction is affected by high indel rates.     

Super pairwise alignment (SPA): an efficient approach to global alignment for homologous sequences.

Sequence analysis is the basis of bioinformatics, while sequence alignment is a fundamental task for sequence analysis. The widely used alignment algorithm, Dynamic Programming, though generating optimal alignment, takes too much time due to its high computation complexity O(N(2)). In order to reduce computation complexity without sacrificing too much accuracy, we have developed a new approach to align two homologous sequences. The new approach presented here, adopting our novel algorithm which combines the methods of probabilistic and combinatorial analysis, reduces the computation complexity to as low as O(N). The computation speed by our program is at least 15 times faster than traditional pairwise alignment algorithms without a loss of much accuracy. We hence named the algorithm Super Pairwise Alignment (SPA). The pairwise alignment execution program based on SPA and the detailed results of the aligned sequences discussed in this article are available upon request.