The space necessary for storing the back-translation graph of a protein sequence P of size n depends linearly on n. Basically, as mentioned in the section dedicated to data structures, the back-translation graph G _P = (V _P , E _P ) consists of 3·n groups of nodes { } (as each of the n amino-acids are encoded by sequences of 3 nucleotides). Every group i contains the nodes corresponding to the nucleotides that can appear at position i in at least one of the putative coding sequences (see (1)). The number of nodes in a group is limited by the number of codons that encode an amino acid, that we denote (6 in the worst case scenario for non-ambiguous symbols) and thus does not depend on the protein's length.

Arcs exist only between nodes in consecutive groups (equation (2)), therefore each node can have a limited number of neighbors. Consequently, the overall memory consumption for storing the back-translation graph of a protein sequence P of size n is (n). The worst case scenario is a protein sequence composed only of the amino acids L, S, R, which are encoded by 6 codons each, and hence have the most complex back-translation. For each such amino acid, 10 nodes and 20 arcs are necessary, yielding a maximum memory size of 30n for the entire graph. For the ambiguous nucleotide symbol encoding though, 6 nodes and 6 arcs are necessary in the worst case for each amino acid, while most amino acids only require 3 nodes and 3 arcs for their back-translated representation.

Let G _A and G _B be graphs obtained by back-translating proteins P _A and P _B , of lengths n _A and n _B respectively. The dynamic programming matrix M computed by the alignment algorithm will have 3·n _A + 1 rows and 3·n _B + 1 columns. Each cell of the matrix M [i, j] has several entries corresponding to the possible pairs of nodes from each sequence. The number of entries is bounded by the square of the number of nodes that can appear on each position in the graph ( ²). Consequently, the total number of entries in the matrix is at most ²·(3·n _A + 1)·(3·n _B + 1), hence (n _A n _B ).

Each entry holds the score of the partial alignment ending at the corresponding positions, as well as the number of frameshifts that occurred on the path so far (to ensure the established limit in the complete alignment) and a reference to the previous matrix entry of the alignment path, to facilitate the traceback. The storage space requirements for this supplementary information are bounded by a constant.

For computing each score in the matrix, the expressions that need to be evaluated are given by equation (3), by querying some of the entries from 5 other cells in the matrix. Since the number of entries in each cell is bounded by ², this operation is considered to be performed in constant time. Consequently, the overall time complexity of the algorithm is (n _A n _B ). To recover the best alignment and the two actual sequences that produce it, a classic traceback algorithm is used, with an execution time depending linearly on the alignment length, which cannot be larger than (3·n _A + 3·n _B + 1).

To overcome the memory issues caused by aligning very large sequences with our dynamic programming method, which requires quadratic space, we used an approach inspired from the linear space algorithm for the LCS problem 26 . Our aim is to decrease the space consumption, not necessarily to linear space, with a less prominent increase of the computation time, i.e. the number of recursive matrix recomputations that are necessary for retrieving the actual alignment in this reduced space.

As a compromise, we choose to split the alignment according to some pre-established cut-points, in submatrices that are small enough to fit into memory, and that are recomputed only once for retrieving the corresponding alignment fragments. In our implementation, the cut-points delimit submatrices that are, by default, 128 columns wide. In this setup, we use a two-step approach: first, we compute the score of the best local alignment in linear space, using a sliding window, while also identifying the intersections of the corresponding path with the established cut-points; in the second step, we recompute separately the submatrices containing parts of the best alignment (restricted to the rows that intersect it), and then rebuild the alignment by pasting the obtained alignment fragments together.

For the first pass, we use a sliding window of 4 columns instead of the original 2, because each partial score depends on the scores that are 3 cells to the left or 3 cells above (see equation (3), items (e) and (f)). Each cell of the sliding window memorizes the matrix entry where the alignment path started (identified by the coordinates within the matrix and actual pair of aligned nodes), as well as the intersections of this path with the cut-points. This information is propagated from the previous cell contributing to the computation of the score, and completed in each cell from the cut-point columns by storing the line number and the node pair that help identify an actual entry in the matrix which belongs to the alignment path. The best scoring entry encountered so far is memorized and updated at each step of the alignment algorithm. When the first pass is completed, the best scoring cell will provide all the necessary information for reconstructing the alignment: the start of the alignment, the intersection with each cut-point, and its end, which is the cell itself. According to these coordinates, subgraphs of the two back-translation graphs are extracted and aligned globally (ensuring that the start and end node pair of each fragment are preserved). The obtained global alignments, combined, will give the best local alignment of the two large sequences.

Once the codon substitution probabilities are obtained, can be deduced in several steps. Basically, we first need to identify all pairs of codons with a common subsequence, that have a perfect semi-global alignment (for instance, codons CAT and ATG satisfy this condition, having the common subsequence AT; this example is further explained below). We then assume that the codons from each pair undergo independent evolution, according to the codon substitution model. For the resulting codons, we compute, based on all possible original codon pairs, p((α _i , p _i , c _i ), (α _j , p _j , c _j )) - the probability that nucleotide α _i , located at position p _i of codon c _i , and nucleotide α _j , situated on position p _j of codon c _j have a common origin (equation (7)). From these, we can immediately compute, as shown by equation (8) below, p((α _i , p _i , a _i ), (α _j , p _j , a _j )), corresponding to the foreground probabilities , where t _i = (α _i , p _i , a _i ) and t _j = (α _j , p _j , a _j ).

In the following, stands for the probability of the event codon c _i mutates into codon c _j in evolutionary time θ, and is given by a codon substitution probability matrix (θ).

The notation c _i [interval _i] ≡ c _j [interval _j] states that codon c _i restricted to the positions given by interval _i is a sequence identical to c _j restricted to interval _j . This is equivalent to having a word w obtained by "merging" the two codons. For instance, if c _i = CAT and c _j = ATG, with their common substring being placed in interval _i = [2..3] and interval _j = [1..2] respectively, w is CATG.

We denote by p(c _i [interval _i] ≡ c _j [interval _j]) the probability to have c _i and c _j , in the relation described above, and we compute it as the probability of the word w obtained by "merging" the two codons. This function should be symmetric, it should depend on the codon distribution, and the probabilities of all the words w of a given length should sum to 1. However, since we consider the case where the same DNA sequence is translated on two different reading frames, one of the two translated sequences would have an atypical composition. Consequently, the probability of a word w is computed as if the sequence had the known codon composition when translated on the reading frame imposed by the first codon, or on the one imposed by the second. This hypothesis can be formalized as:

where (w) and (w) are the probabilities of the word w in the reading frame imposed by the position of the first and second codon, respectively. This is computed as the products of the probabilities of the codons and codon pieces that compose the word w in the established reading frame. In the previous example, the probabilities of w = CATG in the first and second reading frame are:

The values of p((α _i , p _i , c _i ), (α _j , p _j , c _j )) are computed as:

from which obtaining the foreground probabilities is straightforward:

The background probabilities of (t _i , t _j ), , can be simply expressed as the probability of the two symbols appearing independently in the sequences:

Earlier we have shown how to compute the translation dependent scores for non-ambiguous nucleotide symbols. However, as mentioned in the section concerning data structures, we work with ambiguous nucleotide symbols, because their usage improves time and memory consumption while providing the same final results. The scores for ambiguous nucleotide symbol pairs are easily obtained as follows:

where is an ambiguous nucleotide symbol representing the possible nucleotides that can appear on position p _i of the codons that encode the amino acid a _i , and set denotes the set of non-ambiguous nucleotide symbols represented by . Basically, the score of pairing two ambiguous symbols is the maximum over all substitution scores for all pairs of nucleotides from the respective sets.

By using ambiguous symbols, less triplets are formed for each amino acid when compared with the non-ambiguous symbol case. 17 amino acids can be anti-translated as tri-mers with just one ambiguous symbol per position, while the others have two alternatives each of the three positions. Therefore, there are 69 different triplets with ambiguity codes to be paired (as opposed to 99), which means more than twice less storage space necessary for the score matrix.

For the reconstruction of the non-ambiguous putative DNA sequences at traceback, the actual pair of nucleotides that have the highest substitution score from the sets corresponding to two paired ambiguous symbols is required. These are easily obtained for each pair of ambiguous symbols as

In this section we have presented a general framework that helps to compute a translation dependent scoring function for DNA sequence pairs, parametrized by a codon substitution model and an evolutionary time measured in expected number of mutations per codon. We consider that the sequences evolve independently, and the distance is relative to the original sequence.

The score significance is estimated according to the Gumbel distribution, where the parameters λ and K are computed with the method described in 32 33 . In the future, we aim at improving our estimation by using a computation method more suited for gapped alignments, such as 34 .

We use two different score evaluation parameter sets for the forward alignment (where the two back-translated graphs that are aligned have the same translation sense) and the reverse complementary alignment (where one of the graphs is aligned with the reverse complementary of the other), because these are two independent cases with different score distributions.

In order to obtain a more refined evaluation of the alignments, we introduce (λ, K) parameters for estimating the score significance of alignment fragments inside which the reading frame difference is preserved. Therefore, there are eight (λ, K) parameters that help to evaluate the alignments (four for the forward alignment sense and four for the reverse complementary alignment sense):

• (λ _FW , K _FW ) for the forward sense and (λ _RC , K _RC ) for the reverse complementary sense respectively, that are used for evaluating the score of the whole alignment.

• (λ _+i , K _+i ) for the forward sense and (λ _-i , K _-i ) for the reverse complementary sense respectively, with i ∈ {0, 1, 2} that are used for evaluating the scores of each alignment fragment within which the reading frame difference is preserved. This second evaluation aims at providing a measure of the actual contribution of each such fragment to the score of the alignment.

The parameters (λ _±i , K _±i ) are estimated on alignments restricted to the respective reading frame difference, where further frameshifts are not allowed, while (λ _FW , K _FW ) and (λ _RC , K _RC ) are computed in a more flexible setup, where a limited number of frameshifts is accepted.

In this section we discuss the behavior of the proposed scoring system when aligning protein sequences without a frameshift. Given their construction method, we expect the scores to reflect the amino acid similarities, but also to be influenced by similarities at the DNA level.

To evaluate how our scores, used in non-frameshifted alignments, would relate to the classic scoring systems used by biological sequence comparison methods, we first compute, for each scoring matrix T corresponding to an evolutionary distance θ, the expected score for each amino acid pair, as:

Where

Then, considering each amino acid pair as an observation, we compute the correlation coefficient of these expected scores and the BLOSUM matrices as given by 35 .

We also evaluate the correlation with the expected amino acid pair scores obtained when the sequences are aligned using a classic nucleotide match/mismatch system. The latter expected amino acid pair scores are also obtained as weighted sums of scores, in a manner similar to the one described by equations (12) and (13), where the score for aligning two symbols has one of the three established values for match, transition mutation or transversion mutation. For these classic scores, we used the values +5, -3, -4 in the examples reported below, although we have not noticed any drastic changes when different sets of values are used. The obtained correlation coefficients are reported in Table 1.

They suggest that the obtained translation dependent score matrices, either obtained from mechanistic or empirical codon substitution models, are a compromise between the "fully selective" BLOSUM matrices and the non-selective DNA scores.

On the one hand, the scores obtained using the mechanistic model do not make use of the selective pressure, and for this reason are more likely to be correlated with the classic DNA scores. On the other hand, the scores based on empirical codon substitution models reflect the constraints imposed by the similarity of the amino acids encoded by the codons. Hence, they show a strong correlation with the BLOSUM matrices when used without a frameshift.

Figure 7 displays the alignment of two transposase variants from Yersinia pestis. Both proteins are widely present on the NCBI nr database. The mechanism involved is (most probably) a programmed translational frameshifting since such mechanism has been quite frequently observed in several other transposases from related species, e.g. as in E. coli 36 .

Two β-glucosidase variants from Xylella fastidiosa are aligned on Figure 8 with both variants widely present on the NCBI nr database. Xylella fastidiosa is a plant pathogen transmitted by Cicadellidae insects (Homalodisca vitripennis, Homalodisca liturata) and responsible for phoney disease on the peach tree, leaf scorch on the oleander, and Pierce's disease on grape. The β-glucosidase is usually required by several organisms to consume cellulose.

Interestingly, β-glucosidase frameshifts have already been studied in 37 on several bacteria including such γ-proteobacteria as Erwinia herbicola and Escherichia coli of Enterobacteriales but not directly observed in Xanthomonadales.

Diversification of venom toxins has been studied in 38 for advanced snakes: frameshifts were one of the most significant mechanisms involved in the "evolution of the arsenal" and its diversification toward specialized prey capture, sometime with a loss of neurotoxicity 39 . We thus studied neurotoxins from several higher snakes.

In Figure 9, we show the alignment of two presynaptic neurotoxins from two higher snakes of the Elapidae family (Bungarus candidus and Naja kaouthia). Most of the sites are conserved: the primary metal binding site and the putative hydrophobic channel remain before the frameshift, and only the fourth (and last) part of the catalytic network seems changed. We also noticed that, in the original second sequence, the Cysteine regions are more conserved at the DNA level than other amino acids, even after the frameshift, which is a strong hint of the non randomness of this part of the alignment.

Following the discovered frameshift of Figure 9, we took into consideration the sequences of Bungarus candidus species that were similar to the non-frameshifted presynaptic neurotoxin of Naja kaouthia. An interesting alignment is presented in Figure 10, showing a protein that aligns to it well but not perfectly (at least 4 non synonymous transitions before the frameshift and 1 transversion after): this lets open the potential "duplicated first then frameshifted" origin of the frameshifted protein. This assumption was strongly supported by the alignment of the two corresponding cDNA [GenBank:AY057881.1] and [GenBank:AY057880.1] of two homologous proteins.

Furthermore, Figure 11 displays an alignment of two presynaptic neurotoxins from two higher snakes of the Elapidae family (Laticauda colubrina and Laticauda laticaudata). It shows that the unidentified peptide is in fact an alternative splicing (or frameshifted) variant of the neurotoxin. Note that BLASTP identification of the frameshifted peptide on the NCBI nr database gives high E-values (minimal is of 0.67) and is thus expected to be missed by automatic prediction tools (since most of the features specific to neurotoxin are not present), whereas the non frameshifted peptide fragment, once identified, gives several E-values <10^-10 on NCBI nr.

Platelet-derived growth factor is a potent mitogen for cells of mesenchymal origin. Binding of this growth factor to its affinity receptor elicits a variety of cellular responses. It is released by platelets upon wounding and plays an important role in stimulating adjacent cells to grow and thereby heals the wound. In Figure 12, we show the alignment of the back-translated human and rat platelet-derived growth factor proteins. The two proteins share high similarity at the amino acid level on the subsequences 1-84 and 113-195. The amino acids 85-112 can be easily aligned with a frameshift, as can be seen in Figure 12, while classic protein alignment reveals little similarity in these areas (Figure 13).

It is interesting to notice that this double frameshift (if confirmed) may have little influence on the protein (only the beginning of the receptor binding interface is modified). It is also interesting to notice that both the "inducing" and "correcting" frameshifts are located on two different exons.