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Title: Probabilistic Phylogenetic Inference with Insertions and Deletions}
Article:
Abstract:
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 dnamlε improves accuracy relative to the usual approach of ignoring gaps, while retaining the computational efficiency of the Felsenstein peeling algorithm.
Introduction:
A fundamental task in sequence analysis is calculating the probability of a multiple alignment given a phylogenetic tree relating the sequences and an evolutionary model describing how sequences change over time. This quantity is already at the heart of phylogenetic tree inference by either maximum likelihood[1] or Bayesian approaches[2]–[5]. It is also desirable to integrate evolutionary modeling into the probabilistic models widely used for other sequence analysis problems such as HMMs and SCFGs for homology search and genefinding[6]. However, whereas HMMs, SCFGs, and other sequence analysis models account for insertion and deletion events in their probabilistic framework, many widely used phylogenetic models only account for residue substitution events. The general approach of modeling residue substitution as a continuous-time Markov process was introduced by the Jukes-Cantor model of nucleotide substitution in DNA[7] and the Dayhoff pam model of amino acid substitution in proteins[8]. It has since been extensively developed both for nucleotides[9]–[14] and amino acids[15]–[18], and extended to models of more than a single residue, such as codon to codon substitutions[19],[20] and RNA basepair to basepair substitutions[21]–[23]. Given a substitution model and a tree, one can efficiently calculate the probability of an ungapped multiple alignment using Felsenstein's peeling algorithm[1]. The Felsenstein algorithm scales linearly with the length of the alignment and the number of sequences. Because of its economy, it is the basis of maximum likelihood methods in many practical phylogenetic inference tools, including phylip[24], paup*[25], and others[26]–[33]. The Felsenstein algorithm is readily integrated with other probabilistic models for ungapped alignment analysis, including HMMs[14],[34]–[38] and SCFGs[21]. However, when this approach is applied to gapped multiple sequence alignments, gap characters are typically treated as missing data( an unknown residue), effectively equivalent to ignoring them. A variety of more formal approaches for treating insertions and deletions exist[39]–[49]. The canonical model in this active area of research is the Thorne-Kishino-Felsenstein model( tkf91)[50]. tkf91 treats insertion and deletion events as a continuous-time process governed by explicit insertion and deletion rate parameters, allowing multiple insertions and deletions to accumulate at the same “site” over long times. tkf91 improved, for example, upon methods that parsimoniously assume no more than one change per site per branch, including the pioneering Bishop and Thompson pairwise alignment likelihood model that preceded tkf91[51]. Many extensions of tkf91 have appeared 52–56, including practical applications for pairwise alignment[57],[58] and multiple alignment[59]–[62]. Several approaches based on tkf91 or related models have addressed the problem of simultaneously aligning and inferring the phylogeny of a group of related sequences[63]–[68]. In general all algorithms in this class have difficult time complexities, worst-case exponential in the number of sequences, and at least in the case of parsimony models akin to tkf91, the problem of inferring the optimal insertion and deletion history given a tree has been formally shown to be NP-complete[47]. This complexity is inherent to the problem. Any implementation of evolutionary models that allow insertions and deletions, tkf91-based or otherwise, seeks to make approximations that make calculations tractable. Here we are specifically concerned with the problem of calculating the probability of a given multiple alignment and phylogeny. A tkf91-based approach for this problem[69] used a clever approach of using ungapped columns in the alignment to constrain and subdivide the solution space( a so-called “homology structure”), but( because tkf91 is not invariant under column rearrangement) a sum over all subalignments compatible with a given homology structure is still required. As a result, though time complexity exponential in the number of sequences can be avoided in the average case where not many gaps occur, the approach remains expensive in absolute terms; a Bayesian Markov chain Monte Carlo phylogenetic inference for an alignment of 10 globins was reported to require 3 CPU hours on a 1. 25 GHz G4 Apple Macintosh[69]. Although tkf91 has many desirable and realistic properties as a model of evolution, it would be advantageous to have even more computationally efficient approaches, particularly for problems where a gain in efficiency might outweigh sacrificing some of the realism of the model. An alternative is to use a continuous-time Markov process for insertions and deletions that remains fully compatible with the Felsenstein algorithm[41],[46]. This requires an unrealistic assumption of column independence, but nonetheless, it may be better than ignoring gaps altogether. Here we further explore such models. We propose a non-reversible generative model based on a Markov process that we readily incorporated into an existing phylogenetic inference application, resulting in a gain in its accuracy.
Discussion:
The tkf91 model and McGuire's model are designed to be reversible. Our model is non-reversible with respect to insertions and deletions. Reversibility is a mathematically convenient assumption in phylogenetic inference( allowing inference on unrooted trees, for example), and for residue substitution events, usually seem reasonable. However, in the case of insertion/deletion processes, reversibility seem less easy to justify, if one expects the insertion rate λ and the deletion rate μ to behave as independent parameters. For a model to be reversible with respect to insertions/deletions, obviously constraints must be imposed on the insertion rate λ and deletion rate μ; moreover, the constraints imposed by reversibility can be counterintuitive. Consider what happens in a case of zero deletion rate and a positive insertion rate. For McGuire's model, reversibility requires that the frequency of gaps is a constant( π–), and that it is related to the rates of insertions and deletions by the condition π– = μ/( λ+μ). For the tkf91 model, reversibility requires a length distribution for evolved sequences identical to that of ancestral sequences which is geometric with parameter λ/μ[85]. So, in McGuire's model, imposing μ = 0 automatically implies that the gap frequency is zero, effectively converting the model into a substitution-only model, regardless of the insertion rate. In tkf91, the length distribution breaks down entirely in the μ<λ regime, and gives arbitrarily large joint “probabilities”. Our model remains valid for any arbitrary( positive or zero) values for insertion and deletion rate. Another consequence of imposing reversibility is that the expected frequencies of insertions and deletions in a pairwise alignment must be identical. For McGuire's model, the expected frequencies of insertions and deletions in a pairwise alignment are given by,( 30) where π– is the constant frequency of a gap, and λ+μ = β( 1−π–), where β is one of the two parameters of the F84 substitution model[14]. For the tkf91 model, the expected frequencies of insertions and deletions in a pairwise alignment are given by( see Text S4):( 31) Therefore in tkf91 for small divergence times, the expected frequency of insertions is solely dependent on the deletion rate μ; indeed, Equation 31 does not depend on the insertion rate at all. If one desires insertion and deletion rates to be independent parameters, it is more logical to expect that the observed frequencies of deletions and insertions should be in general different, and should depend on both deletion and insertion rates. This is the case in our model, shown in Equations 16 and 17. At one extreme, for λ = 0( zero insertion rate), and. At another extreme, for λ = ∞,, for any positive time t. However, this is not to say that our model is free of its own problems. Most significantly, our model's assumption of column independence is problematic. Rather, in return for this simplification, our model's advantage is that it allows computationally efficient likelihood inference while using a birth-death generative model allowing arbitrary rates of insertion and deletion.
We describe a computationally efficient method to use insertion and deletion events , in addition to substitutions , in phylogenetic inference . To date , many evolutionary models in probabilistic phylogenetic inference methods have only accounted for substitution events , not for insertions and deletions . As a result , not only do tree inference methods use less sequence information than they could , but also it has remained difficult to integrate phylogenetic modeling into sequence alignment methods ( such as profiles and profile-hidden Markov models ) that inherently require a model of insertion and deletion events . Therefore an important goal in the field has been to develop tractable evolutionary models of insertion/deletion events over time of sufficient accuracy to increase the resolution of phylogenetic inference methods and to increase the power of profile-based sequence homology searches . Our model offers a partial answer to this problem . We show that our model generally improves inference power in both simulated and real data and that it is easily implemented in the framework of standard inference packages with little effect on computational efficiency ( we extended dnaml , in Felsenstein's popular phylip package ) .