We undertook to determine the ability of leading bioinformatics programs, including ProtPal, to characterize mutation event histories. We simulated indel histories on a tree, then attempted to reconstruct the MAP history, , using only knowledge of the sequences and the phylogeny (but not the sequence alignment). The history is the aligned set of observed extant and predicted ancestral sequences, such that insertion, deletion, and substitution events can be pinpointed to specific tree branches (though not to specific time points on those branches).

We then characterized the reconstruction quality both directly, by comparison of to the true , and indirectly, by using to estimate , the evolutionary parameters:(1)where the latter step assumes a flat prior, We then compared the history-conditioned parameter estimate to the true .

This statistic is not without its problems. For one thing, we use an initial guess of to estimate . Furthermore, for an unbiased estimate, we should sum over all histories, rather than conditioning on the MAP reconstructed history. This summing over histories would, however, require multiple expensive calculations of , where conditioning on requires only one such calculation. Furthermore, parameter estimation conditioned on a MAP-reconstructed history is the de facto method employed by large-scale genomics studies focusing on indels [24]–[27].

We present here a comprehensive set of reconstructions accounting for the evolutionary history of individual codons in human genes. We used genes in the Orthologous and Paralogous Transcripts in Clades (OPTIC) database's Amniota set, comprised of the 5 mammals H. sapiens, M. musculus, C. familiaris, M. domestica, O. anatinus and G. gallus as an outgroup [30]. Considering only those families with one unique ortholog per species (approximately 7,500 families), we combined tree branch statistics across genes, using the species tree in Text S1. Our reconstructions are available at http://biowiki.org/oscar/optic_reconstruction.tar, and we provide here various graphical summaries of Amniota evolutionary history. Several negative results stand in contrast to earlier-reported trends.

Our algorithm may be viewed as a generalization of Felsenstein's pruning recursion [19], a widely-used algorithm in bioinformatics and molecular evolution. A few common applications of this algorithm include estimation of substitution rates [37]; reconstruction of phylogenetic trees [38]; identification of conserved (slow-evolving) or recently-adapted (fast-evolving) elements in proteins and DNA [39]; detection of different substitution matrix “signatures” (e.g. purifying vs diversifying selection at synonymous codon positions [40], hydrophobic vs hydrophilic amino acid signatures [41], CpG methylation in genomes [42], or basepair covariation in RNA structures [43]); annotation of structures in genomes [44], [45]; and placement of metagenomic reads on phylogenetic trees [36].

Felsenstein's algorithm computes for a substitution model by tabulating intermediate probability functions of the form , where represents the individual residue state of ancestral node , and represents all the sequence data that is causally descended from node in the tree (i.e. the observed residues at the set of leaf nodes whose most recent common ancestor is node ).

The pruning recursion visits all nodes in postorder. Each function is computed in terms of the functions and of its immediate left and right children (assuming a binary tree):where is the probability that node has state , given that its parent node has state ; and is a Kronecker delta function terminating the recursion at the leaf nodes of the tree. These functions are often referred to as “messages” in the machine-learning literature [46].

Our new algorithm is algebraically equivalent to Felsenstein's algorithm, if the concept of a “substitution matrix” over a particular alphabet is extended to the countably-infinite set of all sequences over that alphabet. Our chosen class of “infinite substitution matrix” is one that has a finite representation: namely, the finite-state transducer, a probabilistic automaton that transforms an input sequence to an output sequence, and a familiar tool of statistical linguistics [47].

By generalizing the idea of matrix multiplication () to two transducers ( and ), and introducing a notation for feeding the same input sequence to two transducers in parallel (), we are able to write Felsenstein's algorithm in a new form (see Section):where is the transducer equivalent of the Kronecker delta . The function is now encapsulated by a transducer “profile” of node .

This representation has complexity for sequences of length , which we reduce to by stochastic approximation of the . This approximation relies on the alignment envelope [48], a data structure introduced by prior work on efficient alignment methods. The alignment envelope is a subset of all the possible histories in which most of the probability mass is concentrated. A related data structure is the partial order graph [21]. Both these data structures can be viewed as ensembles of possible histories, in contrast to a single “best-guess” reconstruction of the history. Figure 5 sampledGraph shows a state graph, with paths through it corresponding to histories relating the two sequences GL and GIV. The paths highlighted in blue form a partial order graph, corresponding to a subset of these histories generated by a stochastic traceback. At each progressive traversal step, we sample a high-probability subset of alignments of two sibling profiles in order to maintain a bound on the state space size. Note that if we sample only the most likely path at every internal node, we essentially recover the progressive algorithm of PRANK, and if we sample and store all solutions, we recover the machine with state space of size .

The definitions and lemmas are presented in a condensed form here, and expanded upon in [49].

A transducer is a tuple where is an input alphabet, is an output alphabet, is a set of states, is the start state, is the end state, is the transition relation, and is the transition weight function.

Suppose that and are transducers.

Let be the product of all transition weights along a state path and let be the sum of such weights for all paths whose input labels, concatenated, yield the string and whose output labels yield .

Equivalence: If and have the same input and output alphabets ( and ) and the same sequence weights , then we say the transducers are equivalent, . Less formally, we will write if .

Moore transducers: The Moore normal form for transducers, named for Moore machines [50], associates input/output with three distinct types of state: Match, Insert and Delete. Paths through Moore transducers can be associated with (gapped) pairwise alignments of input and output sequences. For any transducer , there exists an equivalent Moore-normal form transducer with and .

Composition: If 's output alphabet is the same as 's input alphabet (), there exists a transducer, , that unifies the output of with the input of , such that :(2)If and are in Moore form, then and .

Intersection: If and have the same input alphabets (), there exists a transducer, , that unifies the input of with the input of . The output alphabet is , i.e. a -output symbol (or a gap) aligned with a -output symbol (or a gap).

Let denote the set of all gapped pairwise alignments of sequences and . Transducer has the property that :(3)If and are in Moore form, then and . Paths through are associated with three-way alignments of the input sequence to the two output sequences.

Identity: Let be a transducer that copies input to output unmodified, so .

Exact match: For any sequence , there exists a Moore-form transducer with and , that rejects all input except , such that if , and if . Note that outputs nothing (the empty string).

Chapman-Kolmogorov transducers: A transducer is probabilistic if represents a probability : that is, for any given input string, , it defines a probability measure on output strings, .

Suppose is a function returning a probabilistic transducer of the form , i.e. a transducer whose transition weight depends on an additional time parameter, , and which satisfies the transducer equivalence .

Then gives the finite-time transition probabilities of a homogeneous continuous-time Markov process on the strings , as the above transducer equivalence is a form of the Chapman-Kolmogorov equation.

If the state space of is finite, then this equation describes a renormalization of the composed state space back down to the original state space . So far, only one nontrivial time-dependent transducer is known that solves this equation exactly using a finite number of states: the TKF91 model [51].

We rewrite the evidence, for sequences , tree , and parameters , in the form where denotes the set of sequences observed at leaf nodes, denotes the stochastic evolutionary processes occuring on the branches, and denotes the probabilistic model for the sequence at the root node of the tree.

The root and branch transducers represent an alternative view of the tree and parameters . The root transducer outputs from the equilibrium or other initial distribution of the process. If is a parent-child pair, then is a time-dependent transducer parameterized by the branch length. In practise, the branch transducers need not satisfy the Chapman-Kolmogorov equation for the following constructs to be of use; for example, the might be approximations to true Chapman-Kolmogorov transducers [52].

Let be a transducer outputting sequences sampled from the prior at the phylogenetic root.

Let be a tree node. If is a leaf, define . Otherwise, let denote the left and right child nodes, and define where is a transducer modeling the evolution on the branch leading to .

Diagramatically we can write as (.. (.. (..)) (.. (..)))

The phylogenetic likelihood is then fully described by .

Like , transducer models a probability distribution over output sequences, but accepts only the empty string as an input sequence. This empty input sequence is just a technical formality (transducers must have inputs); if we ignore it, we can think of and as hidden Markov models (HMMs), rather than transducers. is an HMM that generates a single sequence, a multi-sequence HMM that generates the whole set of leaf sequences.

Inference with HMMs often uses a dynamic programming matrix (e.g. the Forward matrix) to track the ways that a given evidential sequence can be produced by a given grammar.

For our purposes it is useful to introduce the evidence in a different way, by transforming the model to incorporate the evidence directly. We augment the state space so that the model is no longer capable of generating any sequences except the observed , by composing 's forked outputs with exact-match transducers that will only accept the observed sequences at the leaves of the tree. This yields a model, , whose state space is of size and, in fact, is directly analogous to the Forward matrix.

If is a leaf node, then let where is the sequence at . Otherwise, .

Diagramatically we can write as (.. (....) (.. ..))

Let . The evidence is .

The net output of is always the empty string. The sequences are recognized as inputs by the transducers at the tips of the tree, but are not passed on as outputs themselves.

Likewise, the input of is the empty string, because accepts only the empty string on its input.

We can think of as a Markov model, rather than an HMM. It has no input or output; rather, the sequences are encoded into its structure.

Transducer has states, which is impractically many, so ProtPal uses a progressive hierarchy of approximations to the corresponding , with state spaces that are bounded in size.

If is a leaf node, let . Otherwise, let where is a subset defined by sampling complete paths through the Markov model and adding the -states used by those paths to , until the pre-specified bound on is reached. Then .

The likelihood of a given history may be calculated by summing over paths through consistent with that history. In the simplest cases (e.g. minimal Moore-form branch transducers), each indel history corresponds to exactly one path, so the MAP indel history corresponds to the maximum-weight state path through .

Let be defined such that it has only one nonzero-weighted pathso a -state is either the start state (), the end state (), a wait state () or a match state (). All these states have the form where represents the number of symbols of that have to be read in order to reach that state, i.e. a “co-ordinate” into . All -states are labeled with such co-ordinates, as are the states of any transducer that is a composition involving , such as or .

For example, in a simple case involving a root node (1) with two children (2,3) whose sequences are constrained to be , the evidence transducer is  = (.. (....) (....))

All states of have the form where , so and similarly for . Thus, each state in is associated with a co-ordinate pair into , as well as a state-type pair .

Let be a node in the tree, let be the set of indices of leaf nodes descended from , and let be the phylogenetic transducer for the subtree rooted at , defined in Section. Let be the state space of .

If is a leaf node descended from , then includes, as a component, the transducer . Any -state, , is a tuple, one element of which is a -state, , where is a co-ordinate (into sequence ) and is a state-type. Define to be the co-ordinate and to be the corresponding state-type.

Let be the function returning the set of absorbing leaf indices for a state, such that the existence of a finite-weight transition implies that for all .

Let be two sibling nodes. The alignment envelope is the set of sibling state-pairs from and that can be aligned. The function indicates membership of the envelope. For example, this basic envelope allows only sibling co-ordinates separated by a distance or less(4)

An alignment envelope can be based on a guide alignment. For leaf nodes and , let be the number of residues of sequence in the section of the guide alignment from the first column, up to and including the column containing residue of sequence .

This envelope excludes a pair of sibling states if they include a homology between residues which is more than from the homology of those characters contained in the guide alignment:(5)

Let be the number of match columns (those columns of the guide alignment in which both and have a non-gap character) between the column containing residue of sequence and the column containing residue of sequence . This envelope excludes a pair of sibling states if they include a homology between residues which is more than matches from the homology of those characters contained in the guide alignment: